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Review of the Real Number System
Chapter 1
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1.1
Basic Concepts
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1.1 Basic Concepts
Objectives
1. Write sets using set notation.
2. Use number lines.
3. Know the common sets of numbers.
4. Find additive inverses.
5. Use absolute value.
6. Use inequality symbols.
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1.1 Basic Concepts
Write Sets Using Set Notation
A set is a collection of objects called the elements, or members, of the set.
• Set braces, { }, are used to enclose the elements.
• For example, 4 is an element of the set, {3, 4, 11, 19}.
• {3, 4, 11, 19} is an example of a finite set since we can count the number of elements in the set.
• A set containing no elements is called the empty set, or the null set, denoted by Ø.
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Examples of Sets
Natural numbers Whole numbers Empty set
N = {1,2,3,4,5,6,...} W = {0,1,2,3,4,5,6,
…} Ø (a set with no
elements)
1.1 Basic Concepts
Certain sets of numbers have names:
Caution:
Ø is the empty set; {Ø} is the set with one element, Ø.
Note: N and W are infinite sets. The three dots, called an ellipsis, mean “continue on in the pattern that has been established.”
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Set-Builder Notation
1.1 Basic Concepts
Sometimes instead of listing the elements of a set, we use a notation called set-builder notation.
{x | x has property P }
such th has a given pthe all eleme set nts ropetf yo a rtxx P
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Listing the Elements in Sets
1.1 Basic Concepts
a) {x | x is a whole number less than 3} The whole numbers less than 3 are 0, 1, and 2. This is the set {0, 1, 2}.
b) {x | x is one of the first five odd whole numbers} = {1, 3, 5, 7, 9}.
c) {z | z is a whole number greater than 11} This is an infinite set written with three dots as {12, 13, 14, 15, … }.
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Using Set-Builder Notation to Describe Sets
a) { 0, 1, 2, 3 } can be described as
{m | m is one of the first four whole numbers}.
b) { 7, 14, 21, 28, … } can be described as
{s | s is a multiple of 7 greater than 0}.
1.1 Basic Concepts
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Using Number Lines
A number line is a way to picture a set
of numbers:
1.1 Basic Concepts
0 5–5 –4 –3 –2 –1 1 2 3 4
0 is neither positive nor negative
Negative numbers Positive numbers
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Using Number Lines
The set of numbers identified on this
number line is the set of integers:
I = {…,–3, –2, –1, 0, 1, 2, 3, …}
1.1 Basic Concepts
0 5–5 –4 –3 –2 –1 1 2 3 4
Each number on the number line is called a coordinate of the point it labels.
Graph of 3
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Rational Numbers 1.1 Basic Concepts
Rational numbers can be expressed as the quotient of two integers, with a denominator that is not 0.
The set of all rational numbers is written:
and are integers, 0 .
pp q q
qRational numbers
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Rational Numbers 1.1 Basic Concepts
Rational numbers can be written in decimal form as:
Terminating decimals:
Repeating decimals:
4 14.8 and 2.8
5 5= =
2 40.66666... 0.66 or 0.363636... 0.36
3 11
Bar means repeating digits.
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Irrational Numbers 1.1 Basic Concepts
Irrational numbers have decimals that neither terminate nor repeat:
3 1.7320508...
11 3.316624...
3.141592...
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Graphs of Rational and Irrational Numbers
11
1.1 Basic Concepts
0 5–5 –4 –3 –2 –1 1 2 3 4
Irrational Numbers
Rational Numbers
p3
0.363
2
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Real Numbers
1.1 Basic Concepts
Rational numbers
Integers
Whole numbers
Natural numbers
Irrational numbers
8, 15, , 4
4 5, - , 0.6, 1.75
9 8
11, 6, 4
0
1, 2, 3, 4,
5, 27, 45
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Relationships Between Sets of Numbers
1.1 Basic Concepts
Real numbers
Irrational numbers
Rational numbers
Integers
Noninteger rational numbers
Positive integers
Zero
Negative integers
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Sets of Numbers
Natural numbers or Whole numbers
Integers
Rational numbers
Irrational numbers
Real numbers
{1, 2, 3, 4, 5, 6, … }
{0, 1, 2, 3, 4, 5, 6, … }
{…,–3, –2, –1, 0, 1, 2, 3, … }
1.1 Basic Concepts
and are integers, 0
pp q q
q
| is represented by a point on the number linex x
| is a real number that is not rationalx x
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Relationships Between Sets of Numbers
Decide whether each statement is true or false.
a) All natural numbers are integers.
b) Zero is an irrational number.
c) Every integer is a rational number.
d) The square root of 9 is an irrational number.
e) is an irrational number.
1.1 Basic Concepts
3
p-
True
False
True
False
False
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Additive Inverse
Additive InverseFor any real number a, the number –a is the additive inverse of a.
1.1 Basic Concepts
–4 units from zero 4 units from zero
0 5–5 –4 –3 –2 –1 1 2 3 4
The number –4 is the additive inverse of 4, and the number 4 is the additive inverse of –4.
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The Minus Sign
The symbol “−” can be used to indicate
any of the following:
1. a negative number, such as –13 or –121;
2. the additive inverse of a number, as in
“ –7 is the additive inverse of 7”.
3. subtraction, as in 19 – 7.
1.1 Basic Concepts
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Signed Numbers / Additive Inverses
• The sum of a number and its additive
inverse is always zero.
4 + (–4) = 0 or –16 + 16 = 0
• For any real number a,
–(–a) = a.
1.1 Basic Concepts
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Absolute Value
Geometrically, the absolute value of a number,
a, written |a| is the distance on the number line
from 0 to a.
1.1 Basic Concepts
Distance is 4,
so |–4| = 4.
Distance is 4,
so |4| = 4.
0 5–5 –4 –3 –2 –1 1 2 3 4
Absolute value is always positive.
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Formal Definition of Absolute Value
if a is positive or zero
if a is negative
aa
a
1.1 Basic Concepts
Evaluate the following absolute value expressions:
|–14| |0| –|9| –|–13|
|14| + |–7| –|6–3| –|8–8|
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Equality vs. Inequality
An equation is a statement that two quantities are
equal.
8 + 3 = 11 19 – 12 = 7
An inequality is a statement that two quantities
are not equal. One must be less than the other.
9 < 12 This means that 9 is less than 12.
–7 > – 10 This means that –7 is greater than –10.
1.1 Basic Concepts
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Inequalities on the Number Line
On the number line, a < b if a is to the left of b;
a > b if a is to the right of b.
0 5–5 –4 –3 –2 –1 1 2 3 4
1.1 Basic Concepts
–2 < 3
1 > –4
The inequality symbol always points to the smaller number.
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Inequality Symbols
1.1 Basic Concepts
Symbol Meaning Example
is not equal to –6 10
is less than –9 –3
is greater than 8 –2
is less than or equal to –8 –8
is greater than or equal to –2 –7
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